Affine moment invariants

The affine transform is general linear transformation of space coordinates of the image:

u = a0 + a1x + a2y
v = b0 + b1x + b2y.

The affine moment invariants are features for pattern recognition computed from moments of objects on images that do not change their value in affine transformation. The geometric moment of order p + q of the image f is defined

.

We can normalize them with respect to translation, then we obtain central moments

,

where xt=m10/m00 and yt=m01/m00 are coordinates of the centroid. The ratios

are then invariant to translation and scaling. There is a few approaches to the further procedure, we can use an area of a triangle and integrate a product of a few triangle areas over the object in a graph method [7,9]. Another approach is based on solution of Cayley - Aronhold differential equation [1,2,11]

.

We can also continue in the normalization and normalize subsequently to first rotation, stretching and second rotation [8], then we obtain normalized moments.

If we use one of the first two methods, we obtain many dependent invariants that must be eliminated. The invariants can be zero, identical, an invariant can be a product of other invariants or their linear combination. If all these dependent invariants are eliminated, we obtain a set of irreducible invariants. The invariants can be also polynomially dependent and their elimination has not been automated yet [10].

An example shows using affine moment invariants for registration of two satellite images:

LANDSAT

SPOT

The satellite images show the same area in Bohemia, north-west from Prague. One of them is Landsat from August 28, 1990, the other is Spot from May 13, 1986, the images were scaled twice down. The Landsat image has resolution 30 m and 7 spectral bands. We computed principal component analysis and used 3 most important components (upper left image) as input for color segmentation by means of region growing. The Spot image has resolution 20 m and its 3 spectral bands were used directly for the segmentation (upper right image). The Spot image was used without correction to skew caused by Earth's rotation to test the affine invariance of the features. A noise was reduced in both images by minimization of the Mumford - Shah functional (middle row of images). Segmented regions are in the bottom row of images, that with a counterpart in the other image are numbered, the matched ones have numbers in a circle. All three methods yielded similar results.